Relativistic molecular symmetry

By incorporating these symmetries in the 4-spinor basis functions, as we have done in our BERTHA code [50-54], we can make substantial computational economies in computing interaction integrals. The angular stracture of Dirac 4-spinors described here is also exploited by the major computer package TSYM, which utilizes projection operators to construct relativistic molecular symmetry orbitals for double valued representations of point groups [77-79]. 

D. Peng, K. Hirao. The symmetrized random matrix approach, an efficient method to obtain relativistic molecular symmetry adapted functions. Theor. Chem. Acc., 129 (2011) 517-525. 

In this chapter, we start with a qualitative introduction of double groups and relativistic molecular symmetries, connecting to the more or less phenomenological introduction this subject is frequently accorded in quantum chemistry. The aim is to provide the necessary insight for those who only need an operational familiarity with double groups. We then turn to a more formal discussion of the symmetry invariance of... 

The ability to use precisely the same basis set parameters in the relativistic and non-relativistic calculations means that the basis set truncation error in either calculation cancels, to an excellent approximation, when we calculate the relativistic energy correction by taking the difference. The cancellation is not exact, because the relativistic calculation contains additional symmetry-types in the small component basis set, but the small-component overlap density of molecular spinors involving basis functions whose origin of coordinates are located at different centres is so small as to be negligible. The non-relativistic molecular structure calculation is, for all practical purposes, a precise counterpoise correction to the four-component relativistic molecular... 

Physical applications An early application of relativistic molecular theory was to heavy atom collisions, and the production of supercritical fields involving highly stripped ions [234-237]. Studies have been made of parity- and time-reversal symmetry violation in diatomic molecules [74,238,239], and of parity violation in small chiral molecules [240-242]. 

The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

Use of group chains to utilize molecular symmetry The relativistic Cl program DIRRCI [36] was designed to calculate RASCI wave functions and was primarily intended for calculations of states of heavy transition metals and rare earth complexes. Since these complexes are often highly symmetric it was beneficial to invest effort in the implementation of a double group symmetry scheme. This scheme will be briefly discussed below because it shows how molecular symmetry can be incorporated in correlated relativistic calculations. 

T. Saue, H. J. A. Jensen. Quaternion symmetry in relativistic molecular calculations The Dirac-Hartree-Fock method. /. Chem. Phys., 111(14) (1999) 6211-6222. 

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

An assembly of molecules, weakly interacting in a condensed phase, has the general features of an oriented gas system, showing spectral properties similar to those of the constitutive molecules, modulated by new collective and cooperative intrinsic phenomena due to the coherent dynamics of the molecular excitations. These phenomena emerge mainly from the resonant interactions of the molecular excitations, which have to obey the lattice symmetry (with edge boundary, dimensionality, internal radiation, and relativistic conditions), with couplings to the phonon field and to the free radiation field. 

An increase in d of the low symmetry molecules and an increase in a within the groups are observed for all the considered compounds. One should note here that ft and a are relativistically decreased due to the relativistic increase in IP, decrease in the molecular size and increase in the covalence of the compounds. 

Molecular geometries of the XH diatomic hydrides (X=Cu, Ag, Au) and XFg hexafluorides (X=S, Se, Mo, Ru, Rh, Te, W, Re, Os, Ir, Pt, U, Np, and Pu) molecules were assumed to be of CooV and Oh symmetry with their bond lengths taken from experiments [28-33]. As the spin function is explicidy included in eq.(l), the Cv and Oh point groups reduce to the CooV and Oh double groups, respectively, in the relativistic DV-Xa calculation. Symmetry orbitals... 

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